A regression-based method for estimating mean ... - Biostatistics

Biostatistics (2000), 1, 3, pp. 299–313 Printed in Great Britain

A regression-based method for estimating mean treatment cost in the presence of right-censoring GEORGE W. CARIDES∗ Merck Research Laboratories, 10 Sentry Parkway, BL 2-3, Blue Bell, PA 19422, USA

george [email protected] JOSEPH F. HEYSE Vaccine and Health Economic Statistics, Merck Research Laboratories, PA, USA

BORIS IGLEWICZ Statistics Department, Temple University, 210 Speakman Hall, Philadelphia, PA 19122, USA S UMMARY In many clinical trials and evaluations using medical care administrative databases it is of interest to estimate not only the survival time of a given treatment modality but also the total associated cost. The most widely used estimator for data subject to censoring is the Kaplan–Meier (KM) or product-limit (PL) estimator. The optimality properties of this estimator applied to time-to-event data (consistency, etc.) under the assumptions of random censorship have been established. However, whenever the relationship between cost and survival time includes an error term to account for random differences among patients’ costs, the dependency between cumulative treatment cost at the time of censoring and at the survival time results in KM giving biased estimates. A similar phenomenon has previously been noted in the context of estimating quality-adjusted survival time. We propose an estimator for mean cost which exploits the underlying relationship between total treatment cost and survival time. The proposed method utilizes either parametric or nonparametric regression to estimate this relationship and is consistent when this relationship is consistently estimated. We then present simulation results which illustrate the gain in finite-sample efficiency when compared with another recently proposed estimator. The methods are then applied to the estimation of mean cost for two studies where right-censoring was present. The first is the heart failure clinical trial Studies of Left Ventricular Dysfunction (SOLVD). The second is a Health Maintenance Organization (HMO) database study of the cost of ulcer treatment. Keywords: Dependent censoring; Efficiency; Heart failure; Quality-adjusted survival; Smoothing methods; Ulcer treatment.

1. I NTRODUCTION 1.1. Background Health care decision makers are requiring information about treatment modalities beyond the traditional measures of safety and efficacy. Foremost among these needs are evaluations of quality of life, ∗ To whom correspondence should be addressed

c Oxford University Press (2000)

G. W. C ARIDES ET AL . + + o + ++ ++ o o + + + o o+o + o o + oo o o + oo + +o+o + o o + + o +o + o + + oo oo+ o+ o + o o o oo +o+o+ + o oooooooo + o o + o o o o oooooo o +o ooo+ ooooo +++ + o + o+o oooo++ ooo oo o o o o + o o o o o + ooo++ oo o+o+ o oo o+ + oo ooooooooo+ +ooo+ ooo+o++ + oo o oo + oo ++o++o+ o o + o o+ + +++++ + o o + oo o o + o o o+ o o++ + + o o + o o o ooooo o oo o o o + o o + o o o o + + o o o+oo +o+++ + o oo oo o oo + o o + + o ooo+++ o o o +o +++ + o o o o o + + + o + o+oo+ oo++++ o + o o + + + o o o oooo o ooo +o++ o + o +oo++ +++++ oo + o o o o + o o o + o o + + o o + o o + + + o +o++++++++ ++ o o +o o + + o o oo o o + + + + oo + o o o + + + +++ ++ o ooo o o o o + o + oo oo o + + + o o oo oo o o o o o o ooo o oooo o+ o o +++++ + +++++ oo + +++ +++ oooooo + ++++ ++ o o o o ++++ ++ ++ o ++++++++ ++ ooo o + o + o o o + o o + ooo + + ++ oo + + o o + o + o + + o o + o ++ o o oooooo + + o o o + o oo o oo o o + o o o o o + + o oo o oo o ++

50000

300

10000 5000

Cumulative Cost ($)

o:dead +:censored

1000

o

o 0.005

0.010

o

o

o

o 0.050

0.100

0.500

1.000

Survival Time (years)

Fig. 1. 3-year cumulative cost versus survival time for enalapril patients (log–log scale).

cost of care, and cost-effectiveness. These initiatives are motivated by many factors, including the growth in the number and proportion of elderly in the U.S. and other countries, and technological progress allowing for better health outcomes at possibly higher prices. A recent consensus report of the Panel on Cost-Effectiveness in Health Care and Medicine (Gold et al., 1996) recommends that analyses address the societal aim of maximizing the years of healthy life gained for its population in return for a given level of investment. Similarly, Medicare has indicated that cost-effectiveness could become a criterion for coverage as is already the case in Australia (Powe and Griffiths, 1995). In this paper we present procedures for estimating the mean cumulative cost of long-term treatment. To illustrate the practical difficulties in estimating mean cost, we introduce a clinical trial example that will be analysed and discussed further in Section 5. The treatment arm of the Studies of Left Ventricular Dysfunction (SOLVD) was a randomized, double-blind, placebo-controlled trial (SOLVD) performed in Belgium, Canada, and the United States to study mortality, hospitalization, and incidence of myocardial infarction (The SOLVD Investigators, 1991). The study population consisted of 2569 patients with symptomatic heart failure who were randomly assigned to receive either enalapril (N = 1285) or placebo (N = 1284) in addition to usual care. Figure 1 shows the plot of 3-year cumulative cost versus survival time (log–log scale) for enalapril patients. Inspection of this plot suggests the existence of two or more strata. These data are also very rightskewed, as is typical for health expenditures: see, for example Powe et al. (1993) and Carides and Heyse (1996). Use of measures insensitive to extreme values, such as the median, are usually not desirable as patients with high resource use are substantial contributors to aggregate treatment costs and therefore need to be taken into account in the analysis. Cost data also tend to be highly variable, requiring larger sample sizes to obtain precise estimates than would be needed for inference about a clinical endpoint (Powe and Griffiths, 1995). A major difficulty with the analysis of cost data is the treatment of censored observations (marked with ‘+’ in Figure 1). On the surface, this difficulty seems to be easily overcome through use of traditional survival analysis methods such as the Kaplan–Meier estimator (KM) (Kaplan and Meier, 1958), but this estimator has been shown to be biased in such situations: see, e.g., Carides and Heyse (1996), Lin et al. (1997) and Carides (1998). Whenever the cumulative cost at the censoring time and the cumulative cost at the time of death are dependent, the censoring is ‘informative’ and the KM estimator is inconsistent. This

A regression-based method for estimating mean treatment cost

301

phenomenon is usually present in cost studies as censored patients with relatively large observed costs will tend to have large costs at the time of death as well. This problem of dependent censoring has also been noted by Gelber et al. (1989) in the context of estimating quality-adjusted survival time. Two published methods now exist which can provide consistent nonparametric estimators for mean cost when censoring is present. One method, proposed by Lin et al. (1997) involves partitioning the time period of interest into intervals. The estimate of mean cost is the sum of the product of the KM probability of death in each interval and the mean total cost from observed deaths in that interval. We will provide details in Section 3. The other method, proposed by Zhao and Tsiatis (1997) for estimating the qualityadjusted survival function, can be modified to estimate the survival distribution for cost. This method would compute the weighted sum of the number of patients with costs exceeding a specific value y, where the weights are inversely proportional to the probability of noncensorship with respect to y. 1.2. Notation We will next introduce the notation, assumptions, and basic models that will be used to describe rightcensored cost data. Let Ti j , i = 1, . . . , N j , j = 1, . . . , k, denote the survival or therapy time of the ith patient in stratum j. To simplify the notation we will consider only one stratum in this section. The Ti are assumed to be iid with survivor function ST (t) = P(T > t). The survival times are censored on the right by the censoring variables Ui , i = 1, . . . , N which are assumed to be iid with survivor function SU (u) = P(U > u), independent of the Ti . Thus for each patient one observes Vi = min(Ti , Ui ) and δi = I (Ti ≤ Ui ),

(1.1)

where δ = 0 denotes a censored observation. Let Yi be the overall treatment cost for patient i, Ii be the initial cost, and Yi∗ = Yi − Ii . Assume that Yi and Ii are independent across patients. We hypothesize that total treatment cost is made up of two components—a deterministic function of survival time, and a random component. That is, Yi∗ = h{g(Ti ), εi }.

(1.2)

The function g is typically nondecreasing in survival time, although nonmonotonic functions are possible. The latter can occur when shorter survival results in extensive hospitalization and, consequently, higher total cost. Simple versions of h are the multiplicative and additive models given, respectively, by Yi∗ = g(Ti )Z j , where Z i = eεi ,

and Yi∗ = g(Ti ) + εi .

(1.3)

The multiplicative model is useful for economic variables such as cost because the standard deviation of the response tends to increase proportionately with the (conditional) mean. Patients with greater survival, and hence longer exposure times, have greater potential for events such as hospitalizations which are expensive. The additive model may be appropriate for milder medical conditions which rarely involve hospitalizations. The censoring of survival times implies that the total costs are themselves subject to censoring. Hence, we have information only on costs accumulated up to the minimum of the survival and censoring time. Additionally, we may have information as to the cost history, i.e. the timing of the costs, but this is not required for the proposed method. Excluding start-up cost, g(t) = E(Y |T = t) is interpreted as the expected cost for a patient with survival time t. Thus, we can write mean treatment cost as ∞ µ= g(t)|d ST (t)|. (1.4) 0

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In Section 2 we obtain our proposed two-stage estimator of survival cost based on equation (1.4) and discuss some of its properties. We also show how to simplify the calculations for situations where the relationship between expected cost and survival time is monotonic. In Section 3 we present simulation results on the proposed estimator and comparisons with the estimator of Lin et al. (1997). In Section 4 we return to the SOLVD heart failure clinical trial and an ulcer treatment database study to discuss the practical implications of estimating mean treatment costs, illustrating some of the advantages of the proposed estimator. Section 5 provides concluding remarks including some possible alterations and extensions to our methodology. 2. T HE TWO - STAGE ESTIMATOR The proposed method involves two stages. In the first stage, we estimate the deterministic component g(t) = E(Y |T = t), treating survival time as fixed. Many choices exist for estimation of this regression function. If the researcher is confident in a parametric model which is linear in the coefficients (usually after an appropriate transformation such as log), then ordinary linear regression can be used. In applications where there is much uncertainty as to the functional form of g(t), however, misspecification can result in bias and loss of efficiency. In these situations, better results can be obtained by relying more heavily on the data rather than on the researcher’s notions about the structure of the relationship. Thus, nonparametric regression and smoothing splines are often a better choice (Eubank, 1999; Loader, 1999; Simonoff, 1996). We recommend that only the uncensored cumulative costs and survival times be used to estimate the regression function g(t) because the average costs of censored patients are typically different from those who are uncensored at the same follow-up time. For example, in AIDS trials where patients are observed up until the minimum of the time of death or censoring, it is usually the case that patients who died have accumulated more cost than censored patients on treatment for the same length of time (Lancaster and Intrator, 1998). Inclusion of the censored observations to estimate g(t) would result in bias in these situations. In the second stage we weight the estimate of the regression g(t) ˆ function by the KM estimated probability of death at time t. The two stage estimator is then µˆ T S

= I¯ +

VMAX

g(t)|d ˆ Sˆ T (t)|,

(2.1)

0

where I¯ is the mean of the start-up costs, Sˆ T is the KM estimator of the survival function, and VMAX is the longest follow-up time observed. This estimator is consistent for the unrestricted mean (1.4) when g(t) ˆ consistently estimates g(t) and the longest follow-up time is a death time rather than a censoring time. If the longest follow-up time is a censoring time, (2.1) will have some negative bias, i.e. it will tend to underestimate the true mean cost. This bias arises because the KM estimator is undefined beyond time VMAX if it is a censoring time because we would not know when this last survivor would have died if he were not censored. Various authors have suggested ways of estimating S(t) beyond time VMAX : see, for example, Klein and Moeschberger (1997) for both nonparametric and parametric methods, and Gillespie et al. (1992) for a comparison of the bias in alternative versions of the KM estimator. Of course, when VMAX is censored, the regression g(t) will also need to be estimated beyond the range of the available data. Consequently, parametric assumptions would be required for estimating g(t) for t > VMAX , whereas nonparametric regression or spline smoothing would still be applicable for t ≤ VMAX . In many studies the problem of how to estimate the regression and survival functions beyond the longest follow-up time observed does not exist because the objective is to estimate the mean restricted to the time


303

interval [0, L], where L ≤ VMAX . The two-stage estimator of the restricted mean cost is defined as L ˆ (2.2) µˆ T S = I¯ + g(t)|d ˆ Sˆ T (t)| + Y¯V ≥L S(L). 0

The estimators (2.1) and (2.2) can be used to estimate mean cost for any g(t), given it satisfies some smoothness properties. If the researcher is confident that g(t) is monotonically nondecreasing, however, then the estimation method can be simplified by applying the Kaplan-Meier estimator directly to g(t) ˆ as follows: g(V ˆ M AX )

µˆ T S = I¯ +

0

Sˆ g(T ) {g(t)}d ˆ g(t), ˆ

(2.3)

where Sˆ g(t) is the KM estimator for g. ˆ If the objective is to estimate the mean restricted to the time interval [0, L], the estimator assuming monotonicity becomes g(L) ˆ ˆ ¯ ˆ µˆ T S = I + Sg(T ) {g(t)}d ˆ g(t) ˆ − g(L) ˆ ST (L) + Y¯V ≥L Sˆ T (L). (2.4) 0

The proposed two-stage estimators in the four forms (2.1)–(2.4) are consistent provided the function g(t) is consistently estimated. Consistency in the estimation of g(t) is ensured for (2.1) and (2.2) when nonparametric regression or smoothing spline methods are used (Eubank, 1999). When parametric assumptions are made, consistency is not ensured when these assumptions are violated. Conversely, the price paid for using nonparametric rather than parametric regression is a loss of efficiency when the parametric assumptions are met. If cumulative cost and survival time are linearly related, the estimator (2.3) will be asymptotically normal provided the coefficient(s) are consistently estimated. For example, for the model Yi = β0 + β1 Ti + εi , where the errors εi are iid with mean 0 and finite variance, the estimator of the mean cost is µˆ γ = βˆ0 + βˆ1 µˆ T where βˆ0 and βˆ1 are the least-squares estimators of the intercept and slope, respectively, and µˆ T is the KM estimator of mean survival. Asymptotic normality follows from the consistency of the least-squares estimators (Jenrich, 1969), the asymptotic normality of the KM estimator, and Slutsky’s theorem. The asymptotic variance is β12 V A R(µˆ T ), where the variance expression for µˆ T is well known (Lawless, 1995, p. 78). The two-stage estimator is also asymptotically normal for nonlinear cases where cost is nonlinear in known coefficients. For example, for the model Yi = aTir Z i , where the errors Z i , are iid with mean 1 and r > 0 is known, the estimator of mean cost is µˆ γ = aˆ µˆ X , where aˆ is a consistent estimator of a > 0 and µˆ x is the KM estimator of the mean of X = T r . Asymptotic normality again follows by Slutsky’s theorem and the asymptotic normality of the KM estimator. The asymptotic variance is a 2 V A R(µˆ x ), where the variance expression for µˆ x is found by substituting x for t in the expression for the linear case. The asymptotic distribution for nonlinear cases where cost is nonlinear in unknown coefficients is not known. In Section 4 we illustrate in the context of a case study that the extreme right-skewness of cost distributions can render the normality assumption highly suspect for sample sizes considered large enough for approximate normality when investigating clinical endpoints. These sample sizes may not be sufficient for cost studies. Thus, except for very large studies, reliance on asymptotic thory is not recommended regardless of the functional form of g(t) and estimation method used. Instead, we recommend bootstrap methods for inference (Efron and Tibshirani, 1993). 3. S IMULATION STUDIES In this section we investigate the performance of the proposed two-stage estimators and compare these with an estimator recently proposed in Lin et al. (1997). The latter authors propose essentially two estimators for mean cost. One approach assumes knowledge of the cost histories for each patient while the

304


other estimator, like ours, does not. We will refer to the latter estimator as Lin’s estimator and compare it to the two-stage estimator. This estimator is defined by µˆ Lin =

T∗

ˆ − S(t ˆ + 1)} Aˆ i { S(t)

(3.1)

r =0

where Aˆ t is the sample mean cost for patients observed to die in the interval [t, t +1) for t = 0, . . . , T −1. For the last interval [T ∗ , ∞), Aˆ t , is calculated by taking the sample mean of all patients with observed times equal to T ∗ regardless of whether they are censored or uncensored. They show that their estimator is consistent and asymptotically normal when censoring only occurs at the endpoints of the intervals. Some negative bias is introduced when censoring times are continuous. This estimator is strictly applicable only for the time-restricted estimation problem because Aˆ T ∗ would tend to underestimate the true mean cost of patients dying in the last interval in the unrestricted case. This negative bias occurs whenever patients who are uncensored tend to have shorter survival times and hence smaller expected costs than censored patients. Table 1 shows a comparison of the Lin and the two stage estimators (2.2) and (2.4). The parameter of interest is the 10-year cost of treatment where the relationship between cost and survival time is Yi = aTir Z i , where Z i ∼ L N (0, σε )/ exp(σε2 /2); i.e. ln(Yi ) = {ln(a) − σε2 /2} + r ln(Ti ) + εi . The censoring distribution is P(U = 1−) = P(U = 2−) = P(U = 3−) = 0.05, P(U = 4−) = P(U = 6−) = P(U = 10−) = 0.10, P(U = 10) = 0.55. This endpoint censoring ensures that Lin’s estimator is consistent. The overall probability of censoring under these two distributions, P(T > U ), is about 0.30. The sample size is N = 100 and the true mean cost is $12 874, $23 871, and $48 670, for r = 0.2, 0.6, and 1.0, respectively. We applied both versions of the two-stage estimator to this simulated data. The two-stage estimator assuming a parametric model is labeled TS-P and was calculated by applying (2.4) where the parameters a, σε2 , and r were estimated by linear regression applied to the log-transformed cumulative costs and survival times of the uncensored patients. The two-stage estimator which does not make any parametric assumptions on the form of g(t) is labeled TS-NP and was calculated using (2.2) with a local regression fit for g(t) ˆ using the S-plus function loess applied to the log-transformed total costs and survival times of the uncensored patients. This returned an estimate of B = ln{g(t)}−σ 2 /2, where σ 2 is the variance of the error term for the log-transformed model. This variance was then estimated using the squared residuals of the fit. We then obtained our estimate of g(t) as g(t) ˆ = exp( Bˆ + σˆ 2 /2). In terms of bias, both Lin’s estimator and the two-stage estimators performed very well. The relative bias was always less than 1%. The two-stage estimators are more efficient than Lin’s estimator in all cases considered. Greater efficiency is obtained by using TS-P rather than TS-NP. This result is not surprising considering that TS-NP does not make any parametric assumptions on the form of g(t). Here we define the percent relative efficiency by the ratio of the empirical sample variance of Lin’s estimator to that of the respective two- stage estimator multiplied by 100. The MSEs were all lower for the two-stage estimators. The differences are large enough to exclude sampling error. These results suggest that we can improve estimation by exploiting the underlying relationship between total treatment cost and survival time. Table 2 reports the results of applying the two-stage estimator (2.2) and Lin’s estimator (3.1) to the estimation of 10-year mean cost where g(T ) = (aT 2 + bT ), a < 0, b > 0—a non-monotonic relationship where maximum expected cost is attained at −b/2a = 5 years. True 10-year mean cost is $1293, $3879, and $6466 under the three considered choices of (a, b). All other assumptions are the same as for the simulations of Table 1. Lin’s estimator was calculated using the definition (3.1). For the two-stage estimator we estimated g(.) by (a) a cubic smoothing spline using the S-Plus function smooth.spline and (b) a local regression using the S-Plus function loess. In terms of bias all the estimators performed well, although the two-stage estimator with loess fit had


305

Table 1. Comparison of the two-stage estimators with Lin’s estimator—monotonic case Measure Percent relative bias Sample standard error

Estimator

Value of r 0.2

0.6

1.0

Lin

−0.4

−0.2

0.4

TS-P

−0.2

−0.4

−0.8

TS-NP

0.9

0.5

0.4

Lin

2142

4563

11028

TS-P

1821

3804

8753

TS-NP

2034

4317

9823

Square root of

Lin

2143

4563

11030

sample mean

TS-P

1821

3805

8763

squared error

TS-NP

2037

4319

9825

Percent relative efficiecy of TS-P

138

144

159

Percentage relative Efficiency of

111

112

126

TS-NP Note: 30% expected censoring at interval endpoints only; Y = 10, 000T r Z ; T ∼ Exponential with rate 16 ; Z ∼ L N (0, 1)/ exp 12 ; N = 100; 2000 replicates.

lowest bias over all parameters simulated. The sample standard error comparison shows that both twostage estimators exhibit substantially higher efficiency than Lin’s estimator. 4. C LINICAL TRIAL AND DATABASE EXAMPLES 4.1. Clinical trial We return now to the SOLVD heart failure clinical study where our goal is to estimate mean 3-year cost for the enalapril group. Because per patient utilization, rather than costs, were collected as part of the trial, Glick, et al. (1995) estimated costs for nine types of nonfatal and fatal hospitalizations, deaths outside the hospital, ambulatory care, and enalapril therapy. These costs were then applied to the patient-level utilization. The present analysis uses the (Glick et al., 1995) cost data. For comparison purposes, we applied the KM, Lin, and two- stage estimators to this data. Of the 1285 patients in the enalapril treatment arm, 21.9%(n = 282) were censored prior to the end of year 3. The KM estimate was calculated by integrating under the survivor curve obtained by treating the total costs of patients as right-censored survival times. Lin’s estimator was calculated by partitioning the 3-year period into six 6-month intervals and using (3.1) with T ∗ = 3. Figure 2 shows the plot of cumulative cost versus survival time (log–log scale) for uncensored patients. This plot reveals two fairly distinct subgroups. We therefore chose to stratify the analysis for the two-stage method by estimating the relationship between cost and survival time for each subgroup separately. The two chosen strata were patients who, prior to the end of year 3, (A) had at least one hospitalization and

306

G. W. C ARIDES ET AL . Table 2. Comparison of the generalized two-stage estimator with Lin’s estimator—non-monotonic case Measure

Value of (a, b)

Estimator (−100, 1000)

(−300, 3000)

(−500, 5000) 0.7

Percent relative

Lin

1.2

0.5

bias

TS spline

−1.5

−0.7

−0.9

TS loess

−0.2

0.0

−0.3

Sample standard

Lin

265

805

1341

error

TS spline

239

727

1202

TS loess

244

721

1219

Square root of

Lin

265

805

1342

sample mean

TS spline

240

727

1203

squared error

TS loess

244

721

1264

Percent relative efficiency of

123

123

124

118

125

121

TS spline Percent relative efficiency of TS loess

Note: 30% expected censoring at interval endpoints only; Y = (aT 2 + bT ) ·

Z ; T ∼ Exponential with rate 1/6; Z ∼ L N (0, 1)/ exp(1/2); N = 100; 2000 replicates.

(B) had zero hospitalizations. Censored patients were assigned to stratum A if they had at least one hospitalization (n = 537) and to stratum B if they survived to the end of year 3 with no hospitalizations (n = 223). For those patients who were lost to follow-up without having been hospitalized (n = 115), we estimated the conditional probability of being hospitalized by time 3 with pˆ i =

Sˆ X (Ui ) − Sˆ X (3) , Sˆ T (Ui ). Sˆ x (Ui )

(4.1)

where Sˆ X is the KM estimator of the survival function for time-to-first- hospitalization, and Ui is the censoring time of the ith patient. Each of these patients was then randomly assigned to one of the two strata based on a random draw from a Bernoulli distribution with success probability pˆ i . Table 3 shows the assumed models and fits for each stratum. Letting X i = gi (T ) and j = A, B, the within-group estimate of mean cost is computed as xˆ j (3) µˆ j = (4.2) Sˆ xi (xˆ j )d xˆ j − gˆ j (3) Sˆ j (3) + Y¯ j 3 Sˆ j (3), 0

where Y¯ j 3 is the within-stratum sample mean cost for patients surviving to the end of year 3. The overall two-stage parametric estimate of mean cost is µˆ T S = µˆ A [1 − Sˆ x (3)] + µˆ B . Sˆ x (3). As an alternative to the two-stage parametric estimator, we applied the two-stage nonparametric estimator (2.2) with separate loess fits for each stratum.


307

Table 3. Models fits for the two-stage estimate of mean 3-year enalapril cost Assumed model

Estimate of E(Y |T = t)

A

Yi = exp{exp(β0 + β1 Ti + εi )}

Yˆ = exp{exp(2.18 + 0.0371t)}

B

Yi = β0 + β1 Ti + εi

Yˆ = 952 + 756t

Stratum

10000 5000

Cumulative Cost ($)

50000

A A A A AA A A AA A A AA AA A A A A AA AA A AA A A A A AA AAAAA A A A AAAA AAA AAA A A AA A A AA AA A AA AA AA AAA AA A A A A A AA A AA AAAA A AA AAA AA A AAAAA A A A A A A A A A A A A A AAAAAAA A AA A AAA AAAAAAA A A A AA A A A AA AAA A A AA A AA A AAAAAAAAAAAA A A A A A A A AA A AA A A A A AA A A AA A A A AAAAA AA AAA A AA A A A A AA A A A A A AA AA A A A A A A A AA A A A AA AA A A AAA AA A A A AA AA A A A AA A AA A A A A A A AAAA A A B A A B A BBB BB B B B BB B B B B BBB B BB BB B B BBB BBBB BB B B BB B B BBB B BB B B B B B BB B B B BB B BB B B B B BB BB B BB BB BB B B

1000

A

B

B 0.005

0.010

B

B B 0.050

0.100

0.500

1.000

Survival Time (years)

Fig. 2. 3-Year cumulative cost versus survival time for enalapril uncensored patients (log–log scale).

While the arguments in Section 2 are easily applied to derive an asymptotic variance estimator for cases where the parameters for stratum A are known, we assume here that they are unknown and therefore need to be estimated. Numerical studies have shown that while the estimator assuming known nonlinear coefficients performs well for relatively light censoring, it tends to underestimate the true variance when censoring is moderate to heavy. A bootstrap estimate of variance is therefore recommended. Table 4 shows the point estimates and 95% confidence intervals for the four methods. The standard errors and confidence intervals for the two-stage estimators and Lin’s estimator were computed based on 1000 bootstrap resamples. This bootstrap resampling accounts for the uncertainty of the parameters of the function g(t) for the two-stage parametric estimator and for the uncertainty of the functional form of g(t) for the two-stage nonparametric (loess) estimator. Both two-stage estimates and Lin’s estimate are close while the KM estimate is relatively high, illustrating the inherent bias in the KM estimator when cumulative cost at censoring time and at the time of death are correlated. The standard error estimates are somewhat lower for the two-stage estimator, reflecting the simulation results which indicated an improvement in efficiency. 4.2. HMO database study—ulcer treatment To illustrate the value of bootstrap inferences in this problem, we briefly consider a second example with only 116 patients. We used the two-stage estimator to estimate the mean cost of an ulcer treatment for a study that included patients who had not completed therapy by the cutoff date of the follow-up

308

G. W. C ARIDES ET AL . Table 4. Estimates of mean 3-year enalapril cost ($) Estimator

Estimate

Standard error

95% confidence interval

Two-stage parametric

11378

421

(10553, 12203)

Two-stage

11203

429

(10362, 12044)

Lin

11306

451

(10422, 12190)

Kaplan–Meier

14478

676

(13153, 15803)

nonparametric

period. The total cost of care for these patients was therefore censored (10.3%). Patients were followed for a maximum of 3.5 years We were interested in estimating the unrestricted mean cost up to the time of ‘cure’. As previously mentioned (Section 3) Lin’s estimate would generally have negative bias for the unrestricted case. However, if we redefine the parameter of interest as the mean restricted to 3.5 years, the ˆ two problems are equivalent in this case because S(3.5) = 0. For the two-stage method, a cost model linear in therapy time was deemed reasonable and yielded g(t) ˆ = $3367.55 + $32.69t. We then calculated the two-stage estimator as µˆ T S = 3367.55 + 32.69µˆ T , where µˆ T is the KM estimate of mean survival time. We also calculated the two-stage estimator using the method described in the previous section with loess fit. To calculate Lin’s estimate we partitioned the time axis into five 6-month intervals and one 12-month interval. This last interval was lengthened in order to include some uncensored costs necessary for the calculation of Aˆ T ∗ −1 . Table 5 shows the point estimates and 95% confidence intervals for the four methods. Consistent with the SOLVD analysis, the Lin point estimate and the two-stage estimate with assumed linear model are close while the KM estimate is relatively high. Interestingly, the nonparametric two-stage estimate with loess fit is relatively low. Figure 3 displays the scatterplot of cumulative cost and ’survival’ time, the linear fit (dashed) , the loess fit (solid), and the average within-interval costs used in the calculation of Lin’s estimator (dotted). The higher point estimates obtained with the parametric two-stage and Lin estimators are mostly due to the combined influence of patients with very short ’survival’ times who had relatively large cumulative costs and the one patient with cost exceeding $110 000. These observations exert high influence on the coefficients of g(.) for the parametric two-stage estimator. Lin’s estimator is also highly influenced by these observations as the estimates of average cost in the first interval and in which the outlier lies are greatly affected. Loess, being a local regression fitting method, is less sensitive to these observations. Thus, the two-stage estimator utilizing a smoother for estimation of g(.) may be preferred in situations where influential observations are present. The standard errors for the two-stage and Lin’s estimators were again computed based on 1000 bootstrap resamples. The standard error estimates for the two stage estimators were lower than that for the Lin estimator, with the estimate for the two-stage nonparametric estimator considerably lower. 95% percent confidence intervals were computed using two different methods. The first method uses the bootstrap estimate of standard error in the computation of the standard normal theory confidence interval. The second type of confidence interval uses the bias-corrected and accelerated (BCa) method (Efron and Tibshirani, 1993). The normal theory confidence intervals can be very inexact (i.e. the actual coverage probability can be far from the stated coverage probability) in the presence of non-normality. The BCa intervals have been shown to be nearly exact and second-order accurate, and are currently recommended for general use


309

Table 5. Estimates of mean ulcer treatment cost ($) Estimator

Estimate

Standard error

95% Normal interval

95% BCa interval

11635

1956

(7801, 15469)

(8979, 17738)

Two-stage nonparametric

9553

1480

(6652, 12454)

(7540, 13587)

Lin

11873

2166

(7628, 16118)

(9000, 18767)

Kaplan–Meier

13051

2131

(8874, 17228)

0

20000

40000

Cumulative Cost ($) 60000 80000

100000

120000

Two-stage Parametric

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Therapy Time (years)

Fig. 3. Cumulative cost versus therapy time for uncensored ulcer patients: - - - linear fit; — loess fit; . . . within-interval means.

(Efron and Tibshirani, 1993, p. 188). The two methods yield very different confidence intervals in this example. Figure 4 shows the histograms and normal probability plots for the bootstrap replicates for the three estimators applied to the ulcer cost data. All three estimators appear to depart from normality. Thus there is evidence that the normality assumption does not hold and the normal-theory confidence intervals are not reliable. 5. D ISCUSSION Estimation of mean cost for medical therapies is of increasing importance. We proposed regressionbased estimators which are consistent under general assumptions. The method is based on the exploitation of the underlying relationship between total treatment cost and survival time. Numerical studies have shown that the estimators perform well for a variety of situations. A particular advantage of our approach is the gain in efficiency over purely nonparametric methods which results from the estimation and use of this relationship.


0

100

200

300

400

310

10000

15000

20000

0

50

100

150

200

250

Replicates -- TS Parametric

6000

8000

10000

12000

14000

0

100

200

300

Replicates -- TS Loess

10000

15000

20000

Replicates -- Lin Estimator

Fig. 4. Histograms and normal probability plots for bootstrap replicates of the three estimators applied to the estimation of mean ulcer treatment cost.

A caveat with respect to the use of the parametric form of the two-stage estimator is the potential for misspecification of the functional form of the relationship or error structure and consequent bias and loss of efficiency. These concerns are not present for the Lin method. We showed how these difficulties can be overcome by estimating the regression function by use of nonparametric regression methods. While not explicitly defined as such in their paper, the Lin et al. estimator can be thought of as a special case of the two-stage estimator defined in (2.2), where g(t) is estimated by a step function rather than a smooth curve. This is the simplest form of nonparametric estimator for a mean function and was originally discussed by Tukey (1961) in the context of exploratory data analysis. The simulation results in our paper indicate that efficiency can be improved by use of smooth functions such as cubic splines or loess fits. Use of the regression-based estimators may also lead to further insights about the mechanism generating the data. Application to the clinical trial example led to stratification and thus a better understanding of the differences between strata. Application to the ulcer database study highlighted the influence of outliers on the final cost estimates. Of course, for applications where the parametric approach to estimation of the regression function is chosen, the usual regression model diagnostics should be used (Meyers, 1990). For nonparametric regression applications the researcher must choose a tuning parameter λ for the


311

estimation of g(t). For the two stage estimators proposed in this paper, λ governs the amount of smoothing with larger values providing more smoothing and smaller values placing a greater premium on goodnessof-fit. For the Lin estimator, λ is interpreted as the number of partitions. Eubank (1999) provides an extensive discussion of several criteria useful for choosing optimal tuning parameters. Most of these methods minimize an estimated expected squared-error loss function which balances the lower variance obtained from large values of λ with the lower bias obtained from smaller values. For local regression estimators such as loess, the choice of λ together with the degree of the local polynomial (usually 1 or 2) determine the degrees of freedom of the fit. The degrees of freedom in this context are a generalization of the number of parameters in a parametric model. See Loader (1999, Chapter 2) for additional details and an illustration. For the simulations shown in this paper, we used the generalized cross-validation criterion for the calculation of the two stage estimator with spline fit. For most of the two stage estimates utilizing loess fits we used the default values for λ (span = 0.75) and polynomial degree (degree = 2) in the S-Plus function loess. The results were insensitive to other choices of λ. For the Lin method the choice of λ must also take into account the nature of the censoring mechanism because the estimator is biased when censoring occurs in the interiors of the intervals. Thus, the optimal number of partitions may be greater than that which is implied by the selection criteria described in Eubank (1999). The two-stage estimators proposed in this paper do not assume any knowledge about the stream of costs over time for individual patients. These cost histories may not be known in some applications. For example, in a retrospective utilization study patients may be asked how many times they were admitted to the hospital and how long their lengths of stay were, but the dates of the hospital stays may not be provided. In those cases where the cost histories are available, some loss of efficiency can result from ignoring this information. Lin et al. (1997) provide an estimator which incorporates these intermediate costs within small intervals, and present simulations which show an improvement in efficiency for some, but not all, cases. The method presented in the paper of Zhao and Tsiatis (1997) requires knowledge of the cost histories but can be modified when this information is not available. Numerical studies have shown that this modified version of the estimator has properties similar to those of the Lin estimator discussed in this paper (Carides, 1998). The two-stage method can be extended to utilize the cost history information if we redefine g(t) to be the expected cost incurred at follow-up time t, where in practice t might be measured in days. This mean function can be estimated using smoothing methods and would be weighted by the KM probability of surviving to the start of day t. The methodology proposed in this paper can also be extended to allow for the effects of both categorical and continuous covariates on mean cost by defining g(t, x) to be the expected cost for a patient with survival time t and covariate vector x. This mean function can be estimated using multivariate smoothing methods (Simonoff, 1996) and would be weighted by an estimate of the conditional survival distribution. It is widely accepted that costs incurred in the future should be discounted relative to those incurred in the present; a dollar spent earlier is equivalent to more than a dollar later. The discount rates used for the US have usually been between 3% and 5% reflecting the rates on savings accounts or certificates of deposit. To account for discounting, the two-stage method can simply model the total discounted cost as a function of survival time. Note that discounting of costs is more important the longer the time horizon and the higher the discount rate. This paper has focused on estimation of mean treatment cost when survival time, and hence total cost, is subject to censoring. The methodology can also be extended to the estimation of the mean for other variables which are related to right-censored failure times, such as quality-adjusted survival time (Carides et al., 1999).

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G. W. C ARIDES ET AL . ACKNOWLEDGEMENTS

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[Received July 7, 1999; revised January 10, 2000; second revision February 21, 2000; accepted for publication February 21, 2000]